lausen Function

Define

			(1)
			(2)

then the Clausen functions are defined by

(3)

sometimes also written as  (Arfken 1985, p. 783).

Then the Clausen function  can be given symbolically in terms of the polylogarithm as

(4)

For , the function takes on the special form

(5)

and for , it becomes Clausen's integral

(6)

The symbolic sums of opposite parity are summable symbolically, and the first few are given by

			(7)
			(8)
			(9)
			(10)
			(11)

for  (Abramowitz and Stegun 1972).

REFERENCES:

Abramowitz, M. and Stegun, I. A. (Eds.). "Clausen's Integral and Related Summations" §27.8 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 1005-1006, 1972.

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, 1985.

Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, pp. 89-90, 2003.

Borwein, J.; Bailey, D.; and Girgensohn, R. Experimentation in Mathematics: Computational Paths to Discovery. Wellesley, MA: A K Peters, p. 27, 2004.

Borwein, J. M.; Broadhurst, D. J.; and Kamnitzer, J. "Central Binomial Sums, Multiple Clausen Values and Zeta Functions." Exp. Math. 10, 25-41, 2001.

Clausen, R. "Über die Zerlegung reeller gebrochener Funktionen." J. reine angew. Math. 8, 298-300, 1832.

Grosjean, C. C. "Formulae Concerning the Computation of the Clausen Integral ." J. Comput. Appl. Math. 11, 331-342, 1984.

Jolley, L. B. W. Summation of Series. London: Chapman, 1925.

Lewin, L. Dilogarithms and Associated Functions. London: Macdonald, pp. 170-180, 1958.

Lewin, L. Polylogarithms and Associated Functions. New York: North-Holland, 1981.

Wheelon, A. D. A Short Table of Summable Series. Report No. SM-14642. Santa Monica, CA: Douglas Aircraft Co., 1953.